In this aritcle by Dobarro and Dozo, the authors give another method calculating the scalar curvatrue on warped products manifolds by making using of conformal change.
Let and be two Riemann manifolds. For on , we consider the warped product
and show the relationship between the scalar curvatures on , on and on . This relationship is a nonlinear partial differential equation satisfied by a power of the weight .
Theorem: Let and denote the scalar curvature on and respectively. Then the following equality holds:
where . Namely,
Proof: Write , so is conformal to on and is conformal to on .
Suppose , we apply Yamabe equation in to obtain that satisfies
with , where denotes the scalar curvature on .
As , we use Yamabe equation in . Hence also satisfies
with where denotes the scalar curvature on and the corresponding laplacian.
From , we deduce that
Working in local coordinates
where and , hence
On the other hand,
By using of this in (3) and multiplying by , we obtain from (4)
From (2) we arrive at
Recalling that and denoting , replacing in terms of in this last equality, then multiplying by , we obtain
For any , , such that
Choosing and in (7), we obtain